I asked earlier how the four premisses of Zeno's argument given in the IEP imply the conclusion that Achilles never reaches the tortoise.
Clearly there are other assumptions that have to be made. There is the ‘and so on’ of premiss 4. But how does that work? Suppose Achilles aims at the exact spot Y where he is going to overtake the tortoise. Clearly when he has reached that spot, he will have reached the tortoise. If he reaches any spot X before that, he will have not reached the tortoise. So all the ‘argument’ appears to be saying, it seems, is that if we take any point X before Y, then there is some distance to go. And if we take any spot X’ between Y and X, there is still some distance to be ‘and so on’. But this ‘and so on’ doesn’t prove anything. It proves simply (or rather, it assumes that) we can take any distance whatever, and cut it somewhere. How does that prove ‘Achilles will never reach the tortoise’?
Graham Priest has a slightly different version of the argument. He says (I paraphrase) In order to get from a to b, you must first get to each point between a and b. But there are infinitely many points between a and b. Hidden premiss: to get to something = to do something. Therefore to get from a to b in a finite time, you must do infinitely many things. But you can’t do infinitely many things in a finite time. Therefore etc. But there is much to challenge there. Is the hidden premiss correct? Is ‘getting to’ a point the same as ‘doing something’? Can we actually ‘get to’ a mathematical point? How? We can cross such a point, of course. But then the argument amounts to this: there are infinitely many collections of finite distances between a and b, and we can traverse any such collection in a finite time. Indeed, clearly we can, for the total length of any such collection will be the length between a and b.
Aristotle mentions the argument several times in the Physics, arguing that we must distinguish the ‘actual’ from the ‘potential’ infinite, but this distinction is not very clear. I don't have the book to hand, so will post something later.